Theorem opprvalg 13194

Description: Value of the opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)

Hypotheses

Ref	Expression
opprval.1	|- B = ( Base ` R )
opprval.2	|- .x. = ( .r ` R )
opprval.3	|- O = (oppr ` R )

Assertion

Ref	Expression
opprvalg	|- ( R e. V -> O = ( R sSet <. ( .r ` ndx ) , tpos .x. >. ) )

Proof of Theorem opprvalg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opprval.3	. 2 |- O = (oppr ` R )
2		df-oppr 13193	. . 3 |- oppr = ( x e. _V |-> ( x sSet <. ( .r ` ndx ) , tpos ( .r ` x ) >. ) )
3		id 19	. . . 4 |- ( x = R -> x = R )
4		fveq2 5515	. . . . . . 7 |- ( x = R -> ( .r ` x ) = ( .r ` R ) )
5		opprval.2	. . . . . . 7 |- .x. = ( .r ` R )
6	4, 5	eqtr4di 2228	. . . . . 6 |- ( x = R -> ( .r ` x ) = .x. )
7	6	tposeqd 6248	. . . . 5 |- ( x = R -> tpos ( .r ` x ) = tpos .x. )
8	7	opeq2d 3785	. . . 4 |- ( x = R -> <. ( .r ` ndx ) , tpos ( .r ` x ) >. = <. ( .r ` ndx ) , tpos .x. >. )
9	3, 8	oveq12d 5892	. . 3 |- ( x = R -> ( x sSet <. ( .r ` ndx ) , tpos ( .r ` x ) >. ) = ( R sSet <. ( .r ` ndx ) , tpos .x. >. ) )
10		elex 2748	. . 3 |- ( R e. V -> R e. _V )
11		mulrslid 12584	. . . . . 6 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
12	11	simpri 113	. . . . 5 |- ( .r ` ndx ) e. NN
13	12	a1i 9	. . . 4 |- ( R e. V -> ( .r ` ndx ) e. NN )
14	11	slotex 12483	. . . . . 6 |- ( R e. V -> ( .r ` R ) e. _V )
15	5, 14	eqeltrid 2264	. . . . 5 |- ( R e. V -> .x. e. _V )
16		tposexg 6258	. . . . 5 |- ( .x. e. _V -> tpos .x. e. _V )
17	15, 16	syl 14	. . . 4 |- ( R e. V -> tpos .x. e. _V )
18		setsex 12488	. . . 4 |- ( ( R e. _V /\ ( .r ` ndx ) e. NN /\ tpos .x. e. _V ) -> ( R sSet <. ( .r ` ndx ) , tpos .x. >. ) e. _V )
19	10, 13, 17, 18	syl3anc 1238	. . 3 |- ( R e. V -> ( R sSet <. ( .r ` ndx ) , tpos .x. >. ) e. _V )
20	2, 9, 10, 19	fvmptd3 5609	. 2 |- ( R e. V -> (oppr ` R ) = ( R sSet <. ( .r ` ndx ) , tpos .x. >. ) )
21	1, 20	eqtrid 2222	1 |- ( R e. V -> O = ( R sSet <. ( .r ` ndx ) , tpos .x. >. ) )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148   _Vcvv 2737   <.cop 3595   ` cfv 5216  (class class class)co 5874  tpos ctpos 6244   NNcn 8917   ndxcnx 12453   sSet csts 12454  Slot cslot 12455   Basecbs 12456   .rcmulr 12531  opp_rcoppr 13192

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1re 7904  ax-addrcl 7907

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-tpos 6245  df-inn 8918  df-2 8976  df-3 8977  df-ndx 12459  df-slot 12460  df-sets 12463  df-mulr 12544  df-oppr 13193

This theorem is referenced by:  opprmulfvalg  13195  opprex  13198  opprsllem  13199